David Cameron has used his final prime minister’s questions to praise the public service of his fellow MPs while roundly mocking his opposite number, Jeremy Corbyn, for being so ineffective that a Conservative voter once urged him to try to keep the Labour leader in his job.

In a session that veered from derision to sentimentality to joking – at one point Cameron held up a photograph of himself with Larry the Downing Street cat, saying it proved they were close – the prime minister received numerous tributes from MPs, but also hit out repeatedly at Corbyn and Labour.

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Cameron, making his last appearance in the House Commons as PM before he makes way for Theresa May later on Wednesday, ended with some mockery at his own expense, turning a jibe he once directed at Tony Blair towards himself.

“The last thing I’d say is you can achieve a lot in politics,” he said to huge cheers. “You can get a lot of things done. And that, in the end – the public service, the national interest – that is what it’s all about. Nothing is really impossible if you put your mind to it. After all, as I once said, I was the future once.”

UK Prime Minister (@Number10gov)

PM: You can achieve a lot of things in politics – that’s what it’s all about. Nothing is really impossible if you put your mind to it. #PMQs

There were numerous tributes to the departing prime minister from fellow MPs, as well as from Corbyn, who wished Cameron and his family well for the future.

Cameron was nonetheless at times quite brutal, especially when he mimicked Corbyn’s practice of reading out emails from concerned voters. The prime minister quoted what he said was a message sent last September in which a Tory voter called Judith urged him to treat the then newly elected Labour leader gently. This was because Corbyn’s deputy, Tom Watson, was “experienced, organised and far more dangerous in the long term”, Cameron read out to cheers.

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Earlier, Cameron had mocked the tortuousness of the opposition’s leadership battle, comparing it unfavourably with the swift coronation of May, after her final challenger, Andrea Leadsom, dropped out on Monday.

“We got on with it – we had resignation, nomination, competition and coronation,” Cameron said. “They haven’t even decided what the rules are yet. If they got into power it would take about a year to work out who would sit there.”

When Corbyn referred to May in a question, Cameron welcomed her accession and made fun of Labour’s lack of any female leaders: “When it comes to women prime ministers, I’m pleased to be able to say that pretty soon it’s going to be 2-0.”

Elsewhere the tone was lighter and more reflective. Cameron began the session with a variant on the normal opening to prime minister’s questions. “This morning I had meetings with ministerial colleagues and others,” he told the Commons. “Other than one meeting this afternoon with Her Majesty the Queen, the diary for the rest of my day is remarkably light.”

The first question was in a similar vein, with the UUP MP Danny Kinahan noting some vacancies Cameron could consider. “I’m told that there are lots of leadership roles out there.” Kinahan said: “There’s the England football team, there’s Top Gear, there’s even across the big pond a role that needs filling.” Cameron replied: “I think I’ll pass.”

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Amid the usual broad run of questions, taking in everything from the Kurds in Iraq to Brexit and the economy, Cameron, watched by his wife, Samantha, and the couple’s children in the public gallery of the Commons, received many tributes.

Corbyn asked a series of questions about homelessness, but, even after the mockery, he was generous in his sendoff, even jokingly asking Cameron to thank his mother for advising him how to dress properly, a reference to a jibe from an earlier question time.

Concluding his final answer, Cameron’s said he would next be watching from the backbenches. “I will miss the roar of the crowd. I will miss the barbs from the opposition. But I will be willing you on,” he said.

He paid tribute to the efforts of all MPs: “People come here with huge passion for the issues they care about. They come here with great love for the constituencies they represent. Yes, we can be pretty tough, and test and challenge our leaders, perhaps more than some other countries. But that is something we should be proud of, and we should keep at it.